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arXiv · 2409.16960

Unified quantitative analysis of the Stokes equations in dilute perforated domains via layer potentials

Wenjia Jing·Yong Lu·Christophe Prange
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Abstract

We develop a unified method to obtain the quantitative homogenization of Stokes systems in periodically perforated domains with no-slip boundary conditions on the perforating holes. The main novelty of our paper is a quantitative analysis of the asymptotic behavior of the two-scale cell correctors via periodic Stokes layer potentials. The two-scale cell correctors were introduced and analyzed qualitatively by Allaire in the early 90's. Thanks to our layer potential approach, we also provide a novel explanation of the conductivity matrix in Darcy's model, of the Brinkman term in Brinkman's model, and explain the special behavior for $d=2$. Finally, we also prove quantitative homogenization error estimates in various regimes of ratios between the size of the perforating holes and the typical distance between holes. In particular we handle a subtle issue in the dilute Darcy regime related to the non-vanishing of the Darcy velocity on the boundary.

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Wenjia Jing, Yong Lu, Christophe Prange. 2024-09-25. Unified quantitative analysis of the Stokes equations in dilute perforated domains via layer potentials. https://arxiv.org/abs/2409.16960

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